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As High Performance Computing (HPC) applications with data security requirements are increasingly moving to execute in the public cloud, there is a demand that the cloud infrastructure for HPC should support privacy and integrity. Incorporating privacy and integrity mechanisms in the communication infrastructure of today's public cloud is challenging because recent advances in the networking infrastructure in data centers have shifted the communication bottleneck from the network links to the network end points and because encryption is computationally intensive. In this work, we consider incorporating encryption to support privacy and integrity in the Message Passing Interface (MPI) library, which is widely used in HPC applications. We empirically study four contemporary cryptographic libraries, OpenSSL, BoringSSL, Libsodium, and CryptoPP using micro-benchmarks and NAS parallel benchmarks to evaluate their overheads for encrypting MPI messages on two different networking technologies, 10Gbps Ethernet and 40Gbps InfiniBand. The results indicate that (1) the performance differs drastically across cryptographic libraries, and (2) effectively supporting privacy and integrity in MPI communications on high speed data center networks is challenging-even with the most efficient cryptographic library, encryption can still introduce very significant overheads in some scenarios such as a single MPI communication operation on InfiniBand, but (3) the overall overhead may not be prohibitive for practical uses since there can be multiple concurrent communications.